Lagrangian reduction of discrete mechanical systems by stages

In this work we introduce a category of discrete Lagrange--Poincare systems LP_d and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete mechanical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in LP_d. We introduce a notion of symmetry groups for objects of LP_d and introduce a reduction procedure that is closed in the category LP_d. Furthermore, under some conditions, we show that the reduction in two steps (first by a closed normal subgroup of the symmetry group and then by the residual symmetry group) is isomorphic in LP_d to the reduction by the full symmetry group.


Introduction
The study of mechanical systems with symmetries is a classical subject. A standard technique used in the area is the construction of a certain dynamical system -the reduced system-where some or all of the original symmetries have been eliminated and whose trajectories can be used to obtain the trajectories of the original system. This general idea has been developed and used in many different contexts. In the Lagrangian formulation of Classical Mechanics, one such approach is given by E. Routh [24], although it was implicit in Lagrange's original ideas. A modern treatment, including nonholonomic constraints, is given by H. Cendra, J. Marsden and T. Ratiu in [4]. In the modern Hamiltonian case, there are the original works of V. Arnold [1], S. Smale [25,26], K. Meyer [22], J. Marsden and A. Weinstein [18] and, among the recent literature, J. Marsden et al. [19]. There is also a field-theoretic version as explained, for instance, by M. Castrillón Lopez and T. Ratiu in [2]. In the case of discrete mechanical systems (DMS), different versions of reduction theory have been considered, among others, by S. Jalnapurkar et al. in [12], R. McLachlan and M. Perlmutter in [21] and the authors in [7]. Reduction theory has also been developed in the context of Lie groupoids and Lie algebroids as discussed by J. C. Marrero et al in [16] and by D. Iglesias et al in [11].
In some cases, if G is a symmetry group of a mechanical system, it may be convenient to consider a partial reduction, that is, the reduction of the system by a subgroup H ⊂ G and, eventually, as a second step, the reduction of any remaining symmetries in the associated reduced system. This process is called reduction by (two) stages. A problem is that, in general, the reduced system associated to a symmetric mechanical system is a dynamical system that is not a mechanical system. Therefore, a second reduction cannot be performed in the framework of mechanical systems. For continuous time systems, the solution found by different authors consists of enlarging the class of systems considered beyond the mechanical ones, developing a reduction theory for those generalized systems that extends the original reduction of mechanical systems and, eventually, considering the reduction by stages in this generalized framework. This is the case, for instance, in the Lagrangian context, of [5], and, for Lagrangian systems with nonholonomic constraints, of [3,6]. In the Hamiltonian case, it is extensively analyzed in [19]. It should be remarked that in the Lie algebroid or Lie groupoid contexts the problem described in this paragraph does not arise as the reduction of an object in one of these categories lies within the same category.
The purpose of the present work is to introduce a generalized framework to study the reduction of DMS by stages. In this sense, it parallels [5] for discrete time mechanical systems. More precisely, a category LP d of discrete Lagrange-Poincaré systems (DLPS) is introduced and it is shown that DMSs are among its objects in a natural way. Also, the reduced systems associated to symmetric DMSs are in LP d . The dynamics of a DLPS is defined via a variational principle that, for a DMS, reduces to the discrete Hamilton Principle. Then, a reduction theory for symmetric DLPSs is developed. It is shown that this theory, when applied to DMS, coincides with the one defined in [7]. At this stage, we can prove the main result of the paper, Theorem 7.6, showing that, under appropriate conditions, the reduction in two stages is feasible and isomorphic in LP d to the full reduction in one step.
The construction of reduced systems considered in [5], [3,6], [15], [7] and here, all require additional data: a connection or a discrete connection on a certain principal bundle. We prove that, even though the reduced DLPSs depend on the specific discrete connection used, any two choices lead to DLPSs that are isomorphic in LP d . Last, we prove that under fairly general conditions discrete connections on the appropriate principal bundles satisfying the conditions required by the reduction by stages results exist.
It is well known and very useful that DMSs carry natural symplectic structures. But, in general, their associated reduced systems are not symplectic; instead, they carry Poisson structures. In contrast, general DLPSs do not carry a natural symplectic or Poisson structure; we prove that, when a Poisson structure is added to a DLPS, then it descends to any reduced system associated to it. A consequence of this fact is that all DLPSs obtained by a finite number of reductions from a symmetric DMS, have a "natural" Poisson structure coming from the symplectic structure of the original DMS.
The plan for the paper is as follows. Section 2 reviews the notion of discrete connection on a principal bundle and some of its basic properties. Section 3 introduces discrete Lagrange-Poincaré systems, their dynamics and explores some examples. Section 4 defines a category whose objects are the DLPSs. Section 5 introduces the notion of symmetry group of a DLPS and, then, constructs a "reduced" DLPS associated to any symmetric DLPS and discrete connection; it also compares the dynamics of the reduced DLPS and that of the original DLPS, proving that the trajectories of one system can be obtained from those of the other. An example of reduction process is analyzed in Section 6. Section 7 considers the reduction of DLPSs in two stages. Section 8 studies some aspects of Poisson structures on DLPSs. The paper closes with Section 9, where we list some basic and general results on group actions on manifolds and on principal bundles; most of this material is standard and it is included to have a unified notation and reference point.
Finally, we wish to thank Hernán Cendra for his continuous interest in this work and many very useful discussions.

General recollections
Let G be a Lie group acting on the left on the manifold Q by l Q in such a way that the quotient map π Q,G : Q → Q/G be a principal G-bundle; we also consider the induced diagonal G-action l Q×Q on Q × Q. Leok, Marsden and Weinstein introduced in [15] a notion of discrete connection on principal G-bundles. The following definition comes from [8], which the reader should refer to for further details on discrete connections. Definition 2.1. Let Hor ⊂ Q × Q be an l Q×Q -invariant submanifold containing the diagonal ∆ Q ⊂ Q × Q. We say that Hor defines the discrete connection A d on the principal bundle π Q,G : Q → Q/G if (id Q × π Q,G )| Hor : Hor → Q × (Q/G) is an injective local diffeomorphism. We denote Hor by Hor A d .
When Hor A d is a discrete connection on π Q,G : Q → Q/G, it is easy to see that for any (q 0 , q 1 ) ∈ Q × Q, there is a unique g ∈ G such that (q 0 , l Q g −1 (q 1 )) ∈ Hor A d , where l Q g (q) := l Q (g, q). In this case, the discrete connection form A d : Q × Q → G is defined by A d (q 0 , q 1 ) := g.
Remark 2.2. It is well known that when the principal G-bundle is not trivial, the existence of g stated above cannot be assured in general. It is true, though, when (q 0 , q 1 ) is in a certain open subset of Q × Q containing the diagonal, known as the domain of the discrete connection. Still, in what follows, we omit this technical detail in order to keep the notation simple.
As in the case of connections on a principal G-bundle, discrete connections define a notion of discrete horizontal lift, that we introduce below. Definition 2.3. Let A d be a discrete connection on the principal G-bundle π Q,G : Q → Q/G. The discrete horizontal lift h d : Q × (Q/G) → Q × Q is the inverse map of the injective local diffeomorphism (id Q × π)| HorA d : is the projection on the second variable. More generally, p j : X 1 × · · · × X k → X j is the projection from the Cartesian product onto the j-th component. Discrete connections and their lifts satisfy a number of properties. The next result reviews some of them.
Proposition 2.5. Let A d be a discrete connection on the principal G-bundle π Q,G : Q → Q/G. Then, (1) the discrete connection form A d and the discrete horizontal lift h d are smooth functions and, (2) if we consider the left G-actions on G and Q × (Q/G) given by (3) More generally, for any g 0 , g 1 ∈ G, for all q 0 , q 1 ∈ Q.
In what follows we use several notions that are reviewed in the Appendix (Section 9). For instance, fiber bundles and their maps are introduced in Definitions 9.6 and 9.13, while the action of a Lie group on a fiber bundle is introduced in Definition 9.14.
When a Lie group G acts on the fiber bundle (E, M, φ, F ) and F 2 is a right G-manifold, it is possible to construct an associated bundle on M/G with total space (E × F 2 )/G and fiber F × F 2 . The special case when F 2 = G acting on itself by r g (h) := g −1 hg is known as the conjugate bundle and is denoted by G E (see Example 9.17).
Proposition 2.6. Let G be a Lie group that acts on the fiber bundle (E, M, φ, F ) and A d be a discrete connection on the principal G-bundle π M,G : M → M/G.
Then, Φ A d and Ψ A d are smooth functions, inverses of each other. If we view E ×M and E × G × (M/G) as fiber bundles over M via φ • p 1 , then Φ A d and Ψ A d are bundle maps (over the identity). In addition, if we consider the left G-actions l E×M and l E×G×(M/G) defined by consisting of manifolds and smooth maps (or bundle maps, understanding that the top row are bundles over M and the bottom row are bundles over M/G). In (2.1), we have defined Lemma 2.8. Let G act on the fiber bundle (E, M, φ, F ) and A d be a discrete connection on the principal G-bundle π M,G : Proof. Since Φ A d is a diffeomorphism and π E×M,G is a surjective submersion, Υ A d is also a surjective submersion. Also, as When G acts on the fiber bundle (E, M, φ, F ) and A d is a discrete connection on the principal G-bundle π M,G : M → M/G, we have the following commutative diagram involving the conjugate bundle G E .

Discrete Lagrange-Poincaré systems
A discrete mechanical system (DMS) as in [17] is a pair (Q, L d ) where Q is a finite dimensional manifold known as the configuration space and L d : Q × Q → R is a smooth function called the discrete lagrangian. Trajectories of such a system are critical points of an action function determined by L d .
In this section we introduce an extended notion of DMS as a dynamical system whose dynamics arises from a variational principle. In addition, we find the corresponding equations of motion. In Section 4, we formulate a categorical framework that contains the extended systems.
3.1. Discrete Lagrange-Poincaré systems and dynamics. The reduction procedure introduced in [7] and reviewed in the unconstrained situation later, in Section 3.2, has a shortcoming in that, in most cases, when applied to a DMS, the resulting dynamical system is not a DMS. The main objective of this paper is to overcome this problem by considering a larger class of discrete mechanical systems that is closed by the reduction procedure. In order to define the larger class of DMSs we will consider more general "discrete velocity" phase spaces than Q × Q; concretely, we will consider spaces of the form E × M , where φ : E → M is a fiber bundle. Furthermore, we will consider discrete time dynamical systems on such spaces, whose dynamics will be defined using a lagrangian function and a variational principle. In this section we study the extended discrete velocity phase spaces and discrete lagrangian systems on them.
The motivation for the notion of extended discrete velocity phase space that we consider in this paper comes from the type of space obtained by the reduction process introduced in [7]. There, the reduced space associated to a discrete system on Q × Q with symmetry group G is the space (Q/G) × (Q/G) × Q/G G, that is a fibered product of the pair bundle (Q/G) × (Q/G) and the fiber bundle G → Q/G, where G = G E for E the fiber bundle id Q : Q → Q (see Example 9.17). This space is not a standard space for a DMS due to the presence of G. Therefore, it seems reasonable to enlarge the class of spaces to be considered by looking at spaces that are the fibered product of a pair bundle M × M and a fiber bundle E → M . In fact, for continuous mechanical systems, this is the approach of [5], where their extended velocity phase space is of the form T Q ⊕ V and V is a vector bundle over Q. Yet, we will consider a minor variation of the preceding idea: instead of (M × M ) × M E, we will consider E × M that, as fiber bundles over M (by φ • p 1 in the second space) are isomorphic. The technical advantage of using this last space is that it is easier to work with a product manifold rather than with a fibered product.
Given a fiber bundle φ : E → M we will denote C ′ (E) := E × M , seen as a fiber bundle over M by φ • p 1 . Similarly, we define the discrete second order manifold Definition 3.3. Let φ : E → M be a fiber bundle. An infinitesimal variation chaining map P on E is a homomorphism of vector bundles over p 1 , according to the following commutative diagram (of vector bundles) where p 1 ((ǫ 0 , m 1 ), (ǫ 1 , m 2 )) := ǫ 0 and p 3 ((ǫ 0 , m 1 ), (ǫ 1 , m 2 )) := ǫ 1 . Notice that since φ : E → M is a fiber bundle, ker(dφ) has constant rank, so it defines a vector subbundle of T E → E.
R is a smooth function and P is an infinitesimal variation chaining map on E.
Definition 3.7. Let M = (E, L d , P) be a DLPS. The discrete action of M is a function from the space of all discrete curves on C ′ (E) to R defined by S d (ǫ · , m · ) := for all infinitesimal variations (δǫ · , δm · ) on (ǫ · , m · ) with fixed endpoints, that is, satisfying (3.1) and (3.2).
The following Proposition characterizes the trajectories of a DLPS in terms of (algebraic) equations.  4) in T * ǫ k E, where D j denotes the restriction to the j-th component of the exterior differential on a Cartesian product.
Next, we introduce sufficient conditions for the existence of a flow on a DLPS M = (E, L d , P). Consider the commutative diagram (of smooth maps) Notice that all trajectories ((ǫ 0 , m 1 Let Z ⊂ T * E be the image of the zero section of the canonical projection T * E → E. It is easy to check that Z ⊂ T * E is an embedded submanifold. Proposition 3.9. Let M, E and Z be as above. ( In addition to what was assumed in part 1, suppose that Proof. Part 1 follows immediately from the transversality argument on page 28 of [9], applied to the point (ǫ 0 , ǫ 1 , m 2 ). Notice that, as (ǫ 0 , ǫ 1 , m 2 ) ∈ E U , it is not the empty set.
Let P : , condition 2i implies that dP (ǫ 0 , ǫ 1 , m 2 ) is an isomorphism and, consequently, P is a local diffeomorphism at (ǫ 0 , ǫ 1 , m 2 ). Hence, there are open sets V 1 ⊂ C ′ (E) and  The following example shows how a DMS can be seen as a DLPS.
Discrete paths (ǫ · , m · ) of M are, in the current context, the same as discrete paths q · in Q 2 . Such discrete paths are trajectories of M if and only if they satisfy (3.4) that, in this case, becomes in [17]) that characterizes the trajectories of (Q, L d ). Hence, all DMSs can be seen as DLPSs whose dynamics coincide with those of the original systems.
Remark 3.13. As, by Example 3.12, all DMSs are DLPSs, we can specialize Proposition 3.9 to the case of a DMS (Q, L d ). A simple analysis provides the following statement. Let (q 0 , q 1 , q 2 ) ∈ Q × Q × Q be a solution of (3.5) (for k = 1) such that L d is regular 3 at (q 0 , q 1 ) and (q 1 , q 2 ). Then there are open sets We emphasize that the existence of a trajectory (q 0 , q 1 , q 2 ) as a staring point cannot be avoided. For example, when Q = R and L d (q 0 , q 1 ) : , there is no trajectory of the form (q 0 , q 1 , q 2 ).
The dynamical system obtained by the reduction process of a symmetric DMS can be seen as a DLPS, as we describe in the following section.

3.2.
Reduced system associated to a symmetric discrete mechanical system. We say that the Lie group G is a symmetry group of the DMS (Q, L d ) if G acts on Q in such a way that the quotient mapping π Q,G : Q → Q/G is a principal G-bundle and L d •l Q×Q g = L d for all g ∈ G. Given such a system we can construct a discrete time dynamical system called the reduced system whose dynamics captures 2 A discrete path x· in a manifold X is an element of the Cartesian product X N , for some N ∈ N. 3 Regularity at (q 0 , q 1 ) means that, with respect to local coordinates q a j (for j = 0, 1 and a = 1, . . . , n := dim(Q)) neat q 0 and q 1 , the matrix the essential behavior of the original dynamics. First we review the construction of the reduced system and, then, compare the dynamics of the reduced to that of the unreduced system. After that, we prove that the reduced system can be seen as a DLPS with the same trajectories.
Given a discrete connection A d on the principal G-bundle π Q,G : Q → Q/G, we can specialize the commutative diagram (2.1) to the case where φ : where G = (Q × G)/G with G acting on Q by l Q and on G by conjugation and, explicitly, The following result from [7] 4 relates the dynamics of the original system to a variational principle for a system on G × (Q/G).
Theorem 3.14. Let G be a symmetry group of the DMS (Q, L d ). Fix a discrete connection A d on the principal G-bundle π Q,G : Q → Q/G. Let q · be a discrete path in Q, r k := π Q,G (q k ), w k := A d (q k , q k+1 ) and v k := π Q×G,G (q k , w k ) be the corresponding discrete paths in Q/G, G and G (see footnote 2). Then, the following statements are equivalent.
Remark 3.15. The more general Theorem 5.11 in [7] requires the additional data of a connection A on the principal G-bundle π Q,G : Q → Q/G. With this additional information the variations δq · are decomposed in A-horizontal and A-vertical parts.
The reduced system associated to (Q, L d ) is the discrete dynamical system on G × (Q/G) whose trajectories are the discrete paths that satisfy the variational principle stated in point 2 of Theorem 3.14. A DLPS M := (E,Ľ d , P) is associated to this reduced system; we prove later that the trajectories of both systems coincide. Define the fiber bundle φ : E → M as the conjugate bundle p Q/G : In order to define the infinitesimal variation chaining function, we consider Υ A d : (3.6). Then define P ∈ hom(p * 3 (T G), ker(dp Q/G )) by Lemma 3.16. Let Q, A d and Υ A d be as before. Then, the following assertions are true.
We skip the proof of Lemma 3.16 as we will be proving more general statements later: see point 2 in Lemma 5.1 for point 1 and Lemma 5.10 for points 2 and 3.
Next, we compare discrete trajectories of M with the reduced trajectories given by part 2 of Theorem 3.14. We denote points in E = G with v and in M = Q/G with r. The following result proves that all discrete paths in C ′ (E) arise from discrete paths in Q.
Proof. See Proposition 5.2, that is the same result, in a more general context.
In what follows, we fix discrete paths (v · , r · ) in M and q · in Q such that The following result compares the infinitesimal variations over (v · , r · ) in M to those coming from (3.7).
Proposition 3.18. With the notation as above, the following statements are true.
(1) Given a fixed endpoint variation δq · over the discrete path q · in Q, the infinitesimal variation (δv · , δr · ) defined by (3.7) is an infinitesimal variation with fixed endpoints over (v · , r · ) in M.
is a trajectory of M if and only if it is a trajectory of the reduced system according to point 2 in Theorem 3.14.
Hence, the family of DLPSs contains in a natural way all DMSs as well as all the dynamical systems obtained by reduction of symmetric DMSs.

Categorical formulation
In many circumstances it is useful to be able to consider "maps" between mechanical systems. One example in the area of interest of this paper is the reduction process, seen as a map from a symmetric system to a reduced one. Another example is the comparison of different reductions of the same symmetric system. More generally, a symmetry could be seen as a map from a system to itself. A common framework for considering spaces together with their maps is provided by constructing a category (see, for instance, [5]). In this section we study the basic properties of DLPSs and their morphisms in this categorical context.
where p A,B j : A → B are the maps induced by the canonical projections of a Cartesian product onto its factors, The following assertions are true.
Proposition 4.4. LP d is a category considering the standard composition of functions and identity mappings.
Proof. In order to prove that the given data defines a category one has to check that the composition mapping is associative and the identities are left and right identities for the composition mapping. The composition of functions and the identity mappings meet those requirements, so the only thing left to prove is that Both properties follow in a lengthy but straightforward manner.
Proof. That F satisfies morphism's conditions 1 and 2 follows easily using the corresponding property of the morphism Υ ′′ to lift the data (point or tangent vector) to C ′ (E) and, then, using Υ ′ to push down to C ′ (E ′ ). Given where the last identity holds because Υ ′′ ∈ mor LP d (M, M ′′ ). Thus, F satisfies morphism's condition 3.
The remaining conditions follow in a similar fashion, and we conclude that The last assertion of the statement follows easily as the first part of the Lemma proves that F −1 is a morphism in LP d and since, as functions, F and F −1 are mutually inverses, they have the same property as morphisms in LP d . The following result exposes the relation between trajectories of a DLPS and their images under a morphism in LP d .
Direct computations using the morphism properties of Υ show that condition (4.4) holds for these variations. Then, using (4.5), where the last equality holds because (δǫ · , δm · ) is an infinitesimal variation with fixed endpoints in M over (ǫ · , m · ), that is a trajectory of M. Finally, as (δǫ ′ · , δm ′ · ) was an arbitrary infinitesimal variation with fixed endpoints in M ′ over the path (ǫ ′ · , m ′ · ), we conclude that (ǫ ′ · , m ′ · ) is a trajectory of M ′ . A similar argument shows that if (ǫ ′ · , m ′ · ) is a trajectory of M ′ , then (ǫ · , m · ) is a trajectory of M.

Reduction of discrete Lagrange-Poincaré systems
The purpose of this section is to define what is meant by a group of symmetries of a DLPS. Also, a reduction result is studied. 5.1. Symmetry groups of discrete Lagrange-Poincaré systems. Recall that a G-action on a fiber bundle consists of a pair of G-actions l E and l M , satisfying a number of conditions (Definition 9.14). We can use these actions to define "diagonal" G-actions on the fiber bundles C ′ (E) and C ′′ (E) by These actions are smooth and free because l E and l M have those properties. In addition, the bundle projection maps of C ′ (E) and C ′′ (E) on M are G-equivariant and π M,G : M → M/G is a principal G-bundle. In fact, it is easy to check that G acts on the fiber bundles φ • p 1 : We can also define G-actions on ker(dφ) and p * 3 (T E) by (5. 2) We denote the G-action on ker(dφ) by l T E because it is the restriction of the natural G-action on T E. The action l T E is well defined by the G-equivariance of φ.
Lemma 5.1. Let G be a Lie group acting on the fiber bundle φ : E → M and A d be a discrete connection on the principal G-bundle π M,G : A d ((ǫ 0 , m 1 ), (ǫ 1 , m 2 )) = ((v 0 , r 1 ), (v 1 , r 2 )). Proof. A simple computation shows that, for ((ǫ 0 , m 1 ), (ǫ 1 , m 2 )) ∈ C ′′ (E), we have Consider the commutative diagram where F E and F GE are the diffeomorphisms introduced in Remark 3.1 and A d is smooth. Furthermore, as the projection to its last two components is simply Υ A d : E × M → G E × (M/G), that is known to be a surjective submersion and applying point 2 to the first component, we conclude that Υ A d is a submersion. We check explicitly that Υ Finally, since the diffeomorphism F E is G-equivariant (when considering the G-actions l C ′′ (E) and l E×E×M ), we conclude that point 3 holds.
Proposition 5.2. Let G be a Lie group acting on the fiber bundle φ : E → M and A d a discrete connection on the principal G-bundle π M,G : M → M/G. Given a discrete path (v · , r · ) = ((v 0 , r 1 Proof. The proof is by induction in the length of the reduced discrete path, N . If N = 0, taking (ǫ 0 , m 1 ) := ( ǫ 0 , m 1 ) solves the problem. Otherwise, assume that the result holds for all lengths < N and (v · , r · ) = ((v 0 , r 1 ), . . . , (v N −1 , r N )). Then, there is a discrete path ((ǫ 0 , m 1 1 , r N )). This proves that ((ǫ 0 , m 1 ), . . . , (ǫ N −2 , m N −1 ), (ǫ N −1 , m N )) is a discrete path in C ′ (E) starting at ( ǫ 0 , m 1 ) and that lifts ((v 0 , r 1 ), . . . , (v N −1 , r N )). This proves that the statement holds for discrete paths of length N so that, by the induction principle, it holds for arbitrary lengths. (1) G acts on the fiber bundle φ : E → M (Definition 9.14), (2) considering the "diagonal action" of G on C ′ (E), l C ′ (E) defined in (5.1), L d is G-invariant, and (3) P is a G-equivariant element of hom(p * 3 (T E), ker(dφ)) for the G-actions l T E and l p * 3 (T E) defined in (5. 2). In other words, Proof. Assume that G is a symmetry group of M. Then, by definition, G acts on the fiber bundle φ : E → M . We have to prove that l is a diffeomorphism, so it has morphism's property 1.
has morphism's property 2. As has morphism's property 3. Also, as on C ′′ (E) we have that we see that l C ′ (E) g has morphism's property 4. As L d • l has morphism's property 5 and Lemma 5.5 shows that morphism's property 6 is valid for l Conversely, if G acts on the fiber bundle φ : E → M and l C ′ (E) g ∈ mor LP d (M, M), the first condition for being a symmetry group is met. The other two follow from morphism's properties 5 and 6, together with Lemma 5.5.
Later on we will be interested in subgroups of a symmetry group of a DLPS. The following results establish that closed subgroups of a symmetry group of a system M are symmetry groups of M.
where ξ E is the infinitesimal generator associated to ξ by the G-action on E. Then, (J d ) ξ (ǫ 0 , m 1 ) := J d (ǫ 0 , m 1 )(ξ). In some sense, these functions resemble the momentum mappings that appear in the context of DMS. It is easy to check that when (ǫ · , m · ) is a trajectory of M, for any ξ ∈ g, 4) for all k = 0, . . . , N − 1. This last expression shows how J d evolves on a given trajectory of M. In particular, when the image of P is contained in ker(D 1 L d ), the momentum is conserved along the trajectories; this is the case of a DLPS arising from a discrete mechanical system (see Example 3.12). Equation (5. 4) can also be compared with the momentum evolution equation in the nonholonomic case: equation (35) in [7].

5.2.
Reduced discrete Lagrange-Poincaré system. Let G be a symmetry group of M = (E, L d , P) ∈ ob LP d . We want to construct a new DLPS that, as will be shown later, will play the role of the reduced system of M. First of all, since G acts on (E, M, φ, F ), the conjugate bundle ( G E , M/G, p M/G , F × G), introduced in Example 9.17, is a fiber bundle.
Direct computations show that the image ofP is contained in ker(dp M/G ).
Notice that this DLPS coincides with the DLPS associated to the reduction of (Q, L d ) in Section 3.2. In other words, the reduced system M/(G, A d ) extends the reduction construction of DMSs introduced in [7]. Proof. We have already noticed that Υ A d : C ′ (E) → G E is a surjective submersion, so that morphism's property 1 holds. By point 2 of Lemma 5.1, morphism's property 2 holds. As is the natural inclusion, we have that D 1 (p 2 • Υ A d ) = dπ M,G • dp 2 • i 1 = 0, as Im(i 1 ) ⊂ ker(dp 2 ), so that morphism's property 3 is valid. As morphism's property 4 is satisfied. Morphism's property 5 is satisfied by G being a symmetry group of M and, by definition ofP (5.5), we see that (4.2) holds, proving that morphism's property 6 holds for Υ A d .
When a DLPS is symmetric, constructing the associated reduced system requires the choice of a discrete connection. The following result proves that all reduced DLPSs obtained from a DLPS by this procedure are isomorphic in LP d , independently of the discrete connection chosen.
Then, we have the following commutative diagrams of smooth maps, where the horizontal arrows are diffeomorphisms (see Proposition 2.6) Joining the two diagrams we obtain the commutative diagram of smooth maps The result then follows from Lemma 4.5 because the horizontal arrow is a diffeomorphism and, by Proposition 5.13, the non-horizontal arrows are morphisms in LP d .

5.3.
Dynamics of the reduced discrete Lagrange-Poincaré system. The following result compares the dynamics of a reduced DLPS to that of the original symmetric system. Proof. As, by Proposition 5.13, , the result follows from Theorem 4.7.
Corollary 5. 16. In the same setting of Theorem 5.15, the following assertions are equivalent.
The following reconstruction result shows how, knowing the discrete trajectories of a reduced system, the trajectories of the original system can be recovered.
Remark 5.18. Theorem 5.17 asserts that all trajectories of a reduced DLPS M /(G, A d ) come from trajectories of the original system M. It is possible to give a direct description of the reconstruction process in terms of lifting discrete paths (see Lemma 5.1 and Proposition 5. 2). This process is inductive, so it suffices to describe the initial step, as we do next.

Example
In this section we illustrate the reduction techniques introduced so far with the reduction of an explicit symmetric DLPS and give a description of the resulting system. 6.1. The system and a symmetry group. The starting point is the DMS (Q, L d ), where Q := C 2 − ∆ xy , for ∆ xy the diagonal in C 2 and where h = 0 is a real constant. This DMS arises as a simple discretization of the mechanical system consisting of two distinct unit-mass particles in the plane that interact via a potential V , that only depends on the distance between the particles. Following Example 3.12, we associate a DLPS M to (Q, L d ). Take the fiber bundle φ : E → M to be id Q : Q → Q, the Lagrangian function L d and P := 0. Define the DLPS M := (Q, L d , P).
Recall that SE (2) is the group of special Euclidean symmetries of R 2 ≃ C. We (2) is a closed normal subgroup that is isomorphic (as a group) to C. SE(2) acts naturally on C by l C (A,v) (z) := Az+v. This action induces the diagonal action of SE(2) on Q×Q by l C 2 (A,v) (q) := (l C (A,v) (q x ), l C (A,v) (q y )) = (Aq x +v, Aq y +v). Since Q is preserved by l C 2 , SE(2) acts smoothly on Q by the restricted action, that we denote by l Q .
It is immediate that l Q is a free action. Being U (1) compact, l Q is a proper action. Then, by Corollary 9.10, π Q,SE (2) : From the previous discussion and the fact that φ = id Q is an SE(2)-invariant trivialization of φ : E → M, we conclude that SE(2) acts on the fiber bundle φ : E → M. As L d •l Q×Q (A,v) = L d for all (A, v) ∈ SE(2) and P := 0 is SE(2)equivariant, we conclude that SE(2) is a symmetry group of M. Being T 2 ⊂ SE(2) a closed subgroup, it is also a symmetry group of M by Proposition 5.8.

6.2.
A discrete connection. In this section we use the canonical real inner products in C 2 and C to produce a discrete connection A T2 d on the principal T 2 -bundle π Q,T2 : Q → Q/T 2 , following the construction given in Section 5 of [8]. The idea of that construction (in the current setting) is as follows. As T 2 acts by isometries on C (with the canonical real product), T 2 acts by isometries on C 2 via the diagonal action (with the canonical real inner product on C 2 ). This last inner product induces a T 2 -invariant riemannian metric on Q. The horizontal subspace for the discrete connection is an open subset of the set of pairs (q 0 , q 1 ) ∈ Q × Q such that q 1 = exp Q (v) for some v ∈ T q0 Q that is orthogonal to T q0 V(q 0 ) = ker(dπ Q,T2 (q 0 )), the tangent space to the l Q -orbit through q 0 .
The previous construction gives that is a discrete connection on the principal T 2 -bundle π Q,T2 : Q → Q/T 2 . Straightforward computations show that the discrete connection form is Remark 6.1. Other discrete connections can be considered on the principal T 2bundle π Q,T2 : Q → Q/T 2 . For instance, one can define an affine discrete connection whose horizontal space consists of a level manifold of the discrete momentum function (see Remark 5. 9), This would lead to the reduction procedure considered in Section 11 of [7] for DMS with horizontal symmetries.
6.3. The reduced system. Using the discrete connection A T2 d we construct the . Below we give an explicit DLPS M ′ , isomorphic to M /(T 2 , A T2 d ).
Clearly Υ is smooth and T 2 -invariant. We intend to define a DLPS structure on φ ′ : E ′ → M ′ in such a way that Υ is a morphism. This forces us to define and P ′ ((r 0 , z 0 ), r 1 ), ((r 1 , z 1 ), 4) where (r, z) ∈ C * × C. A number of computations confirm that M ′ := (E ′ , L d ′ , P ′ ) is a DLPS and that Υ ∈ mor LP d (M, M ′ ).
By the T 2 -invariance of Υ, there is a smooth map Υ such that the diagram As Υ is onto and satisfies Υ −1 (Υ(q 0 , q 1 )) = l Q×Q T2 (q 0 , q 1 ) for all (q 0 , q 1 ) ∈ Q × Q, which easily implies that Υ is one to one,Υ is a diffeomorphism. By Lemma 4.5,Υ is an isomorphism in LP d .
All together, M ′ is an explicit model for the reduced DLPS M /(T 2 , A T2 d ).
It should be noticed that the z k are (proportional to) the velocity of the center of mass of the original system, which explains the fact that z k is constant for a trajectory, while r k gives the position of one particle relative to the other.
Remark 6.2. There is a U (1) action on M ′ given by l E ′ (A,0) (r, z) := (Az, Az). This action is a "residue" of the original SE(2) action on M. This action is, indeed, a symmetry of M ′ and can be reduced using the same techniques. During the rest of the paper we will show that, under appropriate conditions, this second reduction produces a system that is isomorphic to M /SE(2).

Reduction in two stages
Let G be a symmetry group of the DLPS M and H ⊂ G a normal closed subgroup. In this section we apply the reduction theory of M by H and, provided that G/H is a symmetry group of M/H, perform a second reduction. Last, we compare the two step reduction with the reduction M/G performed in one step. ).
Being a product of smooth actions, it is a smooth action. Also, as l E is a free and proper G-action (see Lemma 9.11), the same is true for l E×H . Therefore, by Lemma 9.12, the function l HE given in ( Define the map Lemma 7.2. Let G be a Lie group acting on Q by the action l Q in such a way that π Q,G : Q → Q/G is a principal G-bundle. Assume that H ⊂ G is a closed and normal subgroup and that A d is a discrete connection on the principal H-bundle π Q,H : Q → Q/H whose domain is G-invariant for the diagonal G-action l Q×Q . Then, the following assertions are equivalent. (1) For each g ∈ G and (q 0 , q 1 ) in the domain of A d , Proof. Recall that (q 0 , q 1 ) in the domain of A d is in Hor A d if and only if A d (q 0 , q 1 ) = e. Assume that (7.3) holds, for each g ∈ G. Let (q 0 , q 1 ) ∈ Hor A d . Then, for any g ∈ G, ) ∈ Hor A d so that h := gA d (q 0 , q 1 ) −1 g −1 satisfies (7.4). As the element of G with this property is unique, we conclude that (7.3) holds.

Recall that Υ
Unraveling the definitions and taking (7.3) into account, we have that

Hence, as Υ
Differentiating the first component of (7.5) we see that, 1 ) for a unique δǫ 1 ∈ T ǫ1 E. Then, using (7.6), we obtain which shows that the unique element of T l E g (ǫ1) E that represents dl HE π G,H (g) (v 1 )(δv 1 . Also, notice that using (7.5) we obtain We use this information to computeP • l p * 3 T ( HE ) π G,H (g) . For any g ∈ G, ). (7.8) Using the G-equivariance of P and (7.6), we have and, using (7. 7), Going back to (7.8), we obtaiň showing thatP is G/H-equivariant. Hence, G/H is a symmetry group of M H .
Example 7.4. In Section 6.1 we introduced a DLPS M and saw that SE(2) was one of its symmetry groups. As T 2 ⊂ SE(2) is a closed normal subgroup, it was also a symmetry group of M. A simple verification shows that the discrete connection form A T2 d defined in (6.2) satisfies (7.3) for G := SE(2) and H := T 2 so that, by Proposition 7.3, SE(2)/T 2 is a symmetry group of M /(T 2 , A T2 d ) ≃ M ′ . As SE(2)/T 2 ≃ U (1), we see that this fact is already suggested in Remark 6.2. smooth inverse of F 2 , showing that F 2 is a diffeomorphism. This proves point 3. By definition, the two triangles involving F 1 in diagram (7.9) are commutative.
The commutativity of the three remaining triangles in diagram (7.9) is due to the commutativity of diagram (2.1).
Theorem 7.6. Consider the data given at the beginning of this section. Let F : be defined by the bottom row of diagram (7.9), that is, Then, the following statements are true.
Proof. By Proposition 2.6, Φ A G d and Φ A G/H acts by isometries; this is the content of Theorem 5.2 in [8]. In addition, A H d is required to satisfy either one of the conditions in Lemma 7.2. In this Section we prove that when the total space Q of a principal G-bundle π Q,G : Q → Q/G is equipped with a G-invariant Riemannian metric, it is possible to apply Theorem 5.2 in [8] to construct discrete connections A H d satisfying the conditions in Lemma 7.2 on the principal H-bundle π Q,H : Q → Q/H for any closed and normal subgroup H ⊂ G.
The construction analyzed in Theorem 5.2 in [8] is as follows. When Q is a Riemannian manifold and a Lie group H acts on Q by isometries, the vertical bundle is defined. Then, a function A H d : U → H is constructed as follows. Given (q 0 , q 1 ) ∈ U, there is r ∈ Q/H such that π Q,H (q 0 ), π Q,H (q 1 ) ∈ W r . Let γ : [0, 1] → Q/H be the unique length-minimizing geodesic contained in W r and joining π Q,H (q 0 ) to π Q,H (q 1 ). Let γ be the A H -horizontal lift of γ to Q, starting at q 0 . Finally, let (7.11) A H d (q 0 , q 1 ) := κ q1 ( γ(1), q 1 ), where κ q1 : Q π Q,H (q1) → H is the smooth map defined by κ q1 (l Q h (q 1 ), q 1 ) := h. Theorem 5.2 in [8] asserts that there is a discrete connection A H d on the principal H-bundle π Q,H : Q → Q/H whose domain is U and whose associated discrete form is given by (7.11).
Below, we consider the case where G is a Lie group and H ⊂ G is a closed normal subgroup. G acts on the Riemannian manifold Q by isometries and in such a way such that π Q,G : Q → Q/G is a principal G-bundle. Then, by restricting the G-action to an H-action, H acts by isometries on Q and π Q,H : Q → Q/H is a principal H-bundle. But, still, G/H acts on Q/H by isometries and making π Q/H,G/H : Q/H → (Q/H)/(G/H) a principal G/H-bundle.
Lemma 7.8. Under the previous conditions, there is a collection of open subsets {W r ⊂ Q/H : r ∈ Q/H} that are geodesically convex as above that, in addition, satisfies for all q ∈ Q and g ∈ G.
Proof. In the current context, we have the commutative diagram where all the π-mappings are principal bundles and φ is a diffeomorphism. Let σ be a section of π Q/H,G/H that may be discontinuous, and define σ : Q → Q/H by σ := σ • φ • π Q,G . It is easy to see that σ is G-invariant, that its image intersects each G/H-orbit in Q/H in exactly one point and that, for each q ∈ Q, σ(q) and π Q,H (q) are on the same G/H-orbit.
Since l Q/H π G,H (g) is an isometry in Q/H, the open sets W π Q,H (q) are also geodesically convex. A direct computation shows that the collection {W π Q,H (q) : q ∈ Q} satisfies (7.12).
structure is important both for the theoretical as well as the numerical applications of DMSs. Still, the dynamical system obtained as the reduction of a DMS may not carry a symplectic structure: an obvious reason could be that dim( G × (Q/G)) = 2 dim(Q) − dim(G) could be odd, making it impossible for the reduced space G × (Q/G) to be a symplectic manifold.
The purpose of this section is to show that when a symmetric DLPS has a Poisson structure, in a sense to be defined below, and the symmetry group acts by Poisson maps, then its reduction also carries a Poisson structure and the reduction morphism is a Poisson map. In principle, these structures could be uninterestingfor instance, the trivial Poisson structure is always a Poisson structure for a DLPS. Still, when a DLPS has an interesting structure, as is the case of those DLPSs obtained from DMSs, the natural Poisson structure arising from the symplectic structure is inherited by all reductions, as we see below.
Proof. Being G a symmetry group of M, by Lemma 2.8, is a principal G-bundle. Then, the G-action on C ′ (E) is free and proper and Υ A d is a surjective submersion. As G acts on C ′ (E) by Poisson maps, it follows from Theorem 10.5.1 in [20] that there is a unique Poisson structure {, } C ′ ( GE ) on C ′ ( G E ) such that Υ A d becomes a Poisson map, hence (8.1) holds. By Theorem 5.15, we have the commutative diagram of manifolds and smooth maps: where F M/G is the flow of the reduced system. As Υ A d and F M are Poisson maps, with Υ A d onto, it follows from Lemma 8.3 below that, F M/G is a Poisson map. All together, we have seen that {, } C ′ ( GE ) is a Poisson structure of M/(G, A d ). Proof. As f • φ 1 = φ 2 , a direct computation shows that, for The result follows by noticing that, as φ 1 is onto, φ * 1 is one to one.
Recall that a regular DMS (Q, L d ) carries a natural closed 2-form ω L d that is symplectic in, at least, an open subset of Q × Q containing the diagonal ∆ Q . We have the following result.
Lemma 8.4. Let G be a symmetry group of the regular discrete mechanical system (Q, L d ). Then, the diagonal G-action on Q × Q is symplectic for the symplectic form ω L d .
Proof. See the argument at the beginning of page 375 in [17].
In particular, a DLPS M = (Q, L d , P) that comes from a DMS (Q, L d ) as in Example 3.12, carries a natural Poisson structure {, } Q×Q arising from the symplectic structure ω L d on Q × Q. It is well known that F * M (ω L d ) = ω L d (see Section 1.3.2 in [17]). Hence F M is a Poisson map and, consequently, {, } Q×Q is a Poisson structure of M.
When G is a symmetry group of (Q, L d ), it is a symmetry group of M and, by Lemma 8.4, it acts on C ′ (Q) = Q × Q by Poisson maps for {, } Q×Q . Fixing a discrete connection A d on π Q,G : Q → Q/G, by Proposition 8.2, the reduced system M/(G, A d ) has a natural Poisson structure induced by {, } Q×Q and Υ A d is a Poisson map.
We conclude that all DLPSs obtained from a DMS by a finite number of reductions have natural Poisson structures that make the corresponding reduction morphisms Poisson maps.

Appendix
The purpose of this Appendix is to review some basic definitions and standard results, using a notation that is compatible with the rest of the paper. Sections 9.1 and 9.2 contain well known material. Section 9.3 contains some nonstandard material. Proposition 9.1. Let M be a manifold and G be a Lie group acting on M by l M . Assume that l M has the property that for any convergent sequence (m j ) j∈N in M and sequence (g j ) j∈N in G such that the sequence (l M gj (m j )) j∈N is convergent, there exists a convergent subsequence of (g j ) j∈N . Then l M is a proper action. Conversely, if the action l M is proper, then the property holds.
Proof. See Proposition 9.13 in [14]. Proof. See Theorem 9.16 in [14]. Proof. An application of the local description of submersions. Definition 9.6. A fiber bundle is a quadruple (E, M, φ, F ) where E, M and F are smooth manifolds and φ : E → M is a smooth map such that each m ∈ M has a neighborhood U ⊂ M and a diffeomorphism Φ U : φ −1 (U ) → U × F that makes the following diagram commutative.

t t t t t t t t t U
In this case, E, M and F are called the total space, base space and fiber of the fiber bundle. A pair (U, Φ U ) as above is called a trivializing chart of the bundle. It is convenient to denote a fiber bundle (E, M, φ, F ) by E or φ.
If (E, M, φ, F ) is a fiber bundle, given two of its trivializing charts (U α , Φ α ) and (U β , Φ β ) such that U αβ := U α ∩ U β = ∅, we can write (Φ α • Φ −1 β )(m, f ) = (m, Φ αβ (m)(f )) for all m ∈ U αβ and f ∈ F , for a smooth map Φ αβ : U αβ → Diff(F ) known as a transition function of the bundle. The fiber bundle is called a Gbundle for a Lie group G if there is a right G-action on F denoted by r F such that all transition functions are of the form Φ αβ (m) = r F χ αβ (m) for a family of smooth functions χ αβ : U αβ → G that satisfy χ βγ (m)χ αβ (m) = χ αγ (m) for all m ∈ U α ∩ U β ∩ U γ = ∅. Definition 9.7. Let (E, M, φ, G) be a G-bundle such that G acts on the fiber G by right multiplication. Then, E is called a principal G-bundle over M . Proof. The first part is direct computation. See Lemma 18.3 in [23] for the converse (in the right action case). the right G-action r F1×F2 g := r F1 g × r F2 g on F 1 × F 2 makes the resulting mapping G-equivariant, in the sense of point 3 of Definition 9.14.
Proposition 9.16. Let G be a Lie group that acts on the fiber bundle (E, M, φ, F ). Then φ induces a smooth mapφ : E/G → M/G such that (E/G, M/G,φ, F ) is a fiber bundle.
Proof. Since the G-actions on E and M are free and proper and φ is equivariant, by Theorem 9.2 and Corollary 9.4, we have that E/G and M/G are manifolds, the quotient mappings π E,G : E → E/G and π M,G : M → M/G are smooth submersions andφ is smooth.
An outline of the proof of the local triviality of (E/G, M/G,φ, F ) goes as follows. Since the existence of local trivializations is a local matter, we can assume that π M,G : M → M/G is a trivial G-principal bundle, that is, it is p 1 : R × G → R for some manifold R and the G-action on M is l R×G g (r, g ′ ) := (r, gg ′ ). Similarly, we can assume that φ : E → M is p 1 : (R × G) × F → R × G and the G-action on E is l (R×G)×F g ((r, g ′ ), f ) := ((r, gg ′ ), r F g −1 (f )). Using Corollary 9.4, p 1 induces a mapp 1 : ((R × G) × F )/G → (R × G)/G = R. In addition, define σ : (R × G) × F → R × F by σ(r, g, f ) := (r, r F g (h)). As σ is smooth and G-invariant, it induces a smooth mapσ : ((R × G) × F )/G → R × F . In fact,σ is a diffeomorphism and satisfies p 1 •σ =p 1 . Thus, we have the following commutative diagram showing the (local) triviality of the bundle (E/G, M/G,φ, F ), ending the proof.
Example 9.17. Applying Proposition 9.16 to the setting of Example 9.15 we conclude that if G acts on the fiber bundle (E, M, φ, F 1 ) and F 2 is a right Gmanifold, then ((E × F 2 )/G, M/G,φ • p 1 , F 1 × F 2 ) is a fiber bundle that we call the associated bundle and denote by F 2E . A special case of this construction is the so called conjugate bundle, denoted by G E , that corresponds to the case when F 2 = G and the right action is r F2 g (h) := l G g −1 (h) = g −1 hg. For the conjugate bundle, we define p M/G :=φ • p 1 .
When the Lie group G acts on a manifold Q in such a way that π Q,G : Q → Q/G is a principal G-bundle, (Q, Q, id Q , {pt}) is a fiber bundle with a G-action. The conjugate bundle in this case, G Q coincides with the conjugate bundle p Q/G : G → Q/G considered in Section 3.2 and in [7].